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Multiple View Geometry

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Outline

Chapters 9 and 10 of “Multiple View Geometry in Computer Vision” by Hartley and Zisserman

Three questions:

Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image?

(ii)Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views?

(iii)Scene geometry (structure): Given corresponding image points xi ↔x’i and cameras P, P’, what is the position of (their pre-image) X in space?

p

p

L2

L2

m1

m1

m1

C1

C1

C1

M

M

L1

L1

l1

l1

e1

e1

lT1

l2

e2

e2

Canonical representation:

l2

m2

m2

m2

l2

l2

Fundamental matrix (3x3 rank 2 matrix)

C2

C2

C2

Epipolar Geometry

Underlying structure in set of matches for rigid scenes

Computable from corresponding points

Simplifies matching

Allows to detect wrong matches

Related to calibration

3D reconstruction of cameras and structure

reconstruction problem:

given xi↔x‘i , compute P,P‘ and Xi

for all i

without additional information possible up to projective ambiguity

Outline of reconstruction

Compute F from correspondences

Compute camera matrices from F

Compute 3D point for each pair of corresponding points

computation of F

use x‘iFxi=0 equations, linear in coeff. F

8 points (linear), 7 points (non-linear), 8+ (least-squares)

(more on this next class)

computation of camera matrices

use

triangulation

compute intersection of two backprojected rays

Reconstruction ambiguity: similarity

Reconstruction ambiguity: projective

Terminology

xi↔x‘i

Original scene Xi

Projective, affine, similarity reconstruction

= reconstruction that is identical to original up to

projective, affine, similarity transformation

Literature: Metric and Euclidean reconstruction

= similarity reconstruction

The projective reconstruction theorem

If a set of point correspondences in two views determine thefundamental matrix uniquely, then the scene and cameras may be reconstructed from these correspondences alone, and any two such reconstructions from these correspondences are projectively equivalent

along same ray ofP2, idem for P‘2

two possibilities: X2i=HX1i, or points along baseline

key result:

allows reconstruction from pair of uncalibrated images

Stratified reconstruction

Projective reconstruction

Affine reconstruction

Metric reconstruction

Projective to affine

(if D≠0)

theorem says up to projective transformation,

but projective with fixed p∞ is affine transformation

can be sufficient depending on application,

e.g. mid-point, centroid, parallellism

Translational motion

points at infinity (not necessarily visible) are fixed for a pure translation

 reconstruction of xi↔xi is on p∞

Scene constraints

Parallel lines

parallel lines intersect at infinity

reconstruction of corresponding vanishing point yields

point on plane at infinity

3 sets of parallel lines allow to uniquely determine p∞

remark: in presence of noise determining the intersection of parallel lines is a delicate problem

remark: obtaining vanishing point in one image can be sufficient

Scene constraints

Scene constraints

*

*

projection

constraints

Affine to metric

identify absolute conic

transform so that

then projective transformation relating original and reconstruction is a similarity transformation

in practice, find image of W∞

image w∞back-projects to cone that intersects p∞ in W∞

note that image is independent of particular reconstruction

Affine to metric

given

possible transformation from affine to metric is

proof:

(Cholesky factorization to obtain A)

Orthogonality

vanishing points corresponding to orthogonal directions

vanishing line and vanishing point corresponding

to plane and normal direction

rectangular pixels

square pixels

Known internal parameters

Same camera for all images

same intrinsics  same image of the absolute conic

e.g. moving camera

given sufficient images there is in general only one

conic that projects to the same image in all images,

i.e. the absolute conic

This approach is called self-calibration

transfer of IAC:

provides 4 constraints, one more needed

(in general two solutions)

Direct metric reconstruction using ω

approach 1

calibrated reconstruction

approach 2

compute projective reconstruction

back-project w from both images

intersection defines W∞ and its support plane p∞

(2 lin. eq. in H-1per view,

3 for two views)

Direct reconstruction using ground truth

use control points XEi with know coordinates

to go from projective to metric

(3 lin. eq. in H per point,

H has 15 d.o.f.)

Reconstruction summary

Given two uncalibrated images compute (PM,P‘M,{XMi})

(i.e. within similarity of original scene and cameras)

Algorithm

Compute projective reconstruction (P,P‘,{Xi})

Compute F from xi↔x‘i

Compute P,P‘ from F

Triangulate Xi from xi↔x‘i

Rectify reconstruction from projective to metric

Direct method: compute H from control points

Stratified method:

Affine reconstruction: compute p∞

Metric reconstruction: compute IAC w

Outline

3-D Reconstruction

Fundamental matrix estimation

Chapters 9 and 10 of “Multiple View Geometry in Computer Vision” by Hartley and Zisserman